Rake Receiver

The Rake receiver is a signal processing structure designed for CDMA and DSSS systems that exploits the multipath nature of wireless channels. Instead of treating multipath components as interference (as done in narrowband systems), the Rake receiver collects energy from multiple delayed copies of the transmitted signal and combines them constructively. This multipath diversity combining is a defining feature of WCDMA, IS-95 CDMA, and any system where the signal bandwidth exceeds the coherence bandwidth of the channel.

Core Concept Explanation

In a wideband CDMA system, the chip duration Tc is much smaller than the delay spread of the multipath channel. This means individual multipath components arrive separated by more than one chip period and can be resolved and tracked independently. The Rake receiver uses multiple processing units called **fingers**, each tuned to a different multipath delay. Each finger performs correlation with the PN code at its assigned delay, despreading the corresponding multipath component and producing a despread output.

After despreading, each finger output is a scaled and phase-rotated version of the transmitted data symbol. The phase rotation arises from the channel propagation delay. Each finger applies a **combining weight** to correct for the phase and to scale by an appropriate factor. The weighted outputs from all fingers are then summed at the combiner. This summation coherently adds the signal components (which are in phase after weight correction) while the noise components add incoherently (power sum), resulting in a net improvement in SNR.

The name Rake comes from the visual resemblance of the finger structure to a garden rake. Each tine of the rake grabs one multipath component. The structure exploits what would otherwise be destructive interference or wasted signal energy by treating each multipath copy as an independent diversity branch.

Mathematical Expression

The combining technique used in the Rake receiver is **Maximal Ratio Combining (MRC)**. In MRC, the weight applied to the output of the i-th finger is wi equals the complex conjugate of the channel coefficient for that path, denoted alpha_i star. The magnitude of wi is proportional to the path amplitude, and the phase of wi cancels the channel phase rotation. The combined output is the sum over all L fingers of wi times the i-th finger output.

The output SNR of MRC combining over L paths equals the sum of the individual path SNRs. If each path has SNR gamma_i, the combined SNR gamma_MRC equals the sum of all gamma_i. This is the fundamental diversity gain: MRC extracts the maximum possible SNR from L independent paths. In comparison, Equal Gain Combining (EGC) corrects only phase (not amplitude), giving slightly lower SNR. Selection combining uses only the best finger.

Given:
Rake receiver with L = 3 fingers
Path SNRs: γ1 = 8 dB, γ2 = 5 dB, γ3 = 3 dB

Why this formula applies:
MRC output SNR = sum of all finger SNRs (linear scale).

Formula:
γ_MRC = γ1 + γ2 + γ3 (sum in linear scale)
Convert dB to linear: γ_lin = 10^(γ_dB/10)

Substitution:
γ1_lin = 10^(8/10) = 6.310
γ2_lin = 10^(5/10) = 3.162
γ3_lin = 10^(3/10) = 1.995

Calculation:
γ_MRC_lin = 6.310 + 3.162 + 1.995 = 11.467
γ_MRC_dB = 10 log10(11.467) = 10.595 dB

Final Answer:
MRC combined SNR = 10.60 dB
Compared to best single finger at 8 dB: gain of 2.6 dB through diversity combining.

Practical Understanding

In IS-95 and WCDMA, the mobile station Rake receiver typically uses 3 to 4 fingers, while the base station may track more paths. The chip rate of 1.2288 Mcps (IS-95) corresponds to a chip period of about 814 nanoseconds, which is a resolvable delay of approximately 244 meters in free space. Any multipath component delayed by more than one chip duration (arriving via a path more than 244 m longer) can be independently resolved and tracked by a separate finger.

The **soft handoff** feature of CDMA systems is an extension of the Rake concept across base stations. During cell edge conditions, the mobile simultaneously connects to two or more base stations. The Rake receiver treats signals from different base stations as additional diversity branches, combining them via MRC. This gives macro-diversity gain and eliminates the hard cutover of FDMA/TDMA systems where a dropped connection at handoff causes service interruption.

Exam Tip: The key Rake receiver fact for GATE: MRC output SNR equals the sum of individual finger SNRs in linear scale. Also remember — a multipath component is resolvable only if its delay difference exceeds one chip duration Tc = 1/Rc. For soft handoff, the Rake treats signals from different base stations as separate diversity branches.

Mechanism: Path Resolution and Diversity Order

• Each Rake finger contains a correlator set to one specific multipath delay τi. It despreads and extracts the i-th path's contribution.
• Two paths are resolvable if their delay difference exceeds the chip period Tc = 1/Rc. Wider bandwidth (higher Rc) resolves more paths.
• MRC weights equal the complex conjugate of the path coefficient. This corrects phase and weights stronger paths more heavily.
• Combined SNR under MRC equals the sum of individual finger SNRs. This is the maximum achievable SNR from L branches.
• Soft handoff extends the Rake concept to signals from different base stations, providing macro-diversity and seamless handoff.

Quick Revision

• Rake receiver collects energy from L multipath components using L parallel fingers (correlators at different delays).
• Resolvability condition: delay difference between two paths must exceed Tc = 1/Rc.
• MRC combining: weight wi = αi* (conjugate of path amplitude). Output SNR = sum of all finger SNRs (linear scale).
• MRC is optimal; EGC corrects phase only; Selection combining uses only the best finger.
• IS-95 chip period ≈ 814 ns, resolves paths separated by more than 244 m in path length.
• Soft handoff uses Rake across base stations — a key CDMA feature absent in FDMA/TDMA systems.
• Trap: MRC SNR = sum of linear SNRs, not sum of dB values. Always convert to linear before adding.